Luis, questions of the form "What is known about X?" are almost always unacceptable here, and are likely to be closed. Please add more context to your question, e.g., "I would like to know if there are results in the literature concerning the abundance of such numbers, and conjectural asymptotics are of special interest", or "I would like to know how these numbers might show up in Iwasawa theory".
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S. Carnahan♦Apr 21 '11 at 7:34

There seems not to be an entry in the OEIS about these prime numbers
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Luis H GallardoApr 21 '11 at 8:33

1 Answer
1

Each fixed $n$ is a special case of Schinzel's "Hypothesis H" (also known as the Bateman-Horn conjecture), which conjectures that there are infinitely many pairs of primes $p,q$ related by your equation $p=2q^n-1$. (For $n=1$ this is a special case of the Hardy-Littlewood prime $k$-tuples conjecture.) So it's conjectured that there are infinitely many examples for each positive $n$, but it's not known even whether there are infinitely many among all $n$ together.